2025 USAMO 第 4 题

Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $FAP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

设$H$为锐角三角形$ABC$的垂心，$F$为从$C$到$AB$的高度的底脚，$P$为$H$在$BC$上的镜像。假设三角形 $FAP$ 的外接圆与线 $BC$ 相交于两个不同的点 $X$ 和 $Y$。证明$C$是$XY$的中点。